1-s2.0-S0141029602001001-main

Engineering Structures 24 (2002) 1561–1574www.elsevier.com/locate/engstruct

Simulation of axisymmetric discharging in metallic silos. Analysisof the induced pressure distribution and comparison with different

standards

M.A. Martınez, I. Alfaro, M. Doblare´ ∗

Department of Mechanical Engineering, Centro Polite´cnico Superior, University of Zaragoza, Marı´a de Luna, 3, 50018, Zaragoza, Spain

Received 2 April 2001; received in revised form 26 June 2002; accepted 27 June 2002

Abstract

The main objective of this paper is to present a new contribution to the problem of dynamic continuum simulation of dischargeof cylindrical silos by the Finite Element Method where many attempts have been made in the past by other researchers. We startwith a study of the bulk solid constitutive behaviour, the analysis of the stored-solid to silo-wall contact interaction and with adiscussion of the remeshing and rezoning algorithms needed to appropriately take into account the large displacements and theassociated mesh distortions that appear in this type of problems. Some restrictions of the simulation are due to the axisymmetryof the model and the constitutive assumptions. First of all, a static analysis is accomplished using the usual hypotheses includedin different standards to check the ability of the method to reproduce standard available solutions. After this calibration stage, adynamic analysis is carried out to take into account the effects induced by the silo quaking phenomena, computing the overpressurefactor and comparing the obtained results with the pressure estimations established by different standards like the European standardEurocode ENV 1991-4, the French AFNOR P22 630, the German DIN 1055 or the American ACI 313-97 and R313-97. 2002Elsevier Science Ltd. All rights reserved.

Keywords:Metallic silos; Dynamic pressure distribution; Discharging processes in silos; Bulk solid materials; Finite element simulation; Rezoningand remeshing

1. Introduction

In spite of considerable experience in the constructionof metallic and concrete silos, their design still lacks ofa global theory generally accepted by researchers. Thisaffirmative sentence is confirmed by the important dif-ferences between the proposals included in the standardsof several countries that, in some cases, lead to very dif-ferent designs.

One of the most important problems for a silodesigner is the accurate prediction of the external loaddistribution acting on the shell, with special care on thewall pressures induced by the stored material. Thispressure distribution depends on the bulk solid constitut-ive behaviour, the interaction between the stored solid

∗ Corresponding author. Tel.:+34-976-761912; fax:+34-976-762578.

E-mail address:[email protected] (M. Doblare´).

0141-0296/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.PII: S0141-0296 (02)00100-1

and the silo walls and the flow properties during the fill-ing and discharging processes [1].

Bulk solids are composed by individual solid particlesinside a continuous phase, usually gaseous. The interac-tion among these particles and the continuous phase iscomplex, being very difficult to formulate a completeand accurate theoretical description of this problem.

This behaviour is a kind of combination betweenliquids and solids. A liquid under static conditions can-not transmit shear forces so its pressure increases lin-early with depth, independent of the direction. Bulk sol-ids, on the contrary, can form surfaces up to a certainslope, corresponding to the natural angle of frictionalstability. They are able to transmit static shear forces andthe pressures on the silo wall do not increase linearlywith depth, but they quickly reach a maximum, due tothe wall friction forces (Fig. 1) [2,3,4–7]. In addition,these pressures depend on the direction and vary accord-ing to whether the solids are being filled, stored or dis-charged. Bulk solids cannot be considered as solid either,

1562 M.A. Martınez et al. / Engineering Structures 24 (2002) 1561–1574

Fig. 1. Different pressure distributions of a liquid and a bulk solid.

since they are not able to carry significant vertical loadswithout lateral support, more like fluids.

The bulk solid within a silo is subjected to two mainpressure states, that, using usual terms in soil mechanics,may be named as active and passive stress states [8,9].The active state develops during filling and remains untilmaterial is removed from the silo. It is identified by thefact that the maximum principal stress s1 at the axis ofsymmetry is the vertical compression stress component.

As soon as the bulk solid starts to flow towards theoutlet, it deforms plastically compressing horizontallyand expanding vertically. This is the typical situation ofa passive stress state that mainly appears in the area ofconvergent flow, in which the major principal stress s1

at the axis of symmetry is now directed horizontally.Pressures acting on the silo walls during discharge

depend very much on the way the convergent flow chan-nel is formed. If it approximately coincides with the hop-per geometry, the whole contents of the silo enters inmotion, exhibiting a flow pattern known as ‘mass flow’ ,in contrast to the so-called ‘ funnel flow’ , where the areaof convergent flow, although well extended into the ver-tical part of the silo, is bounded by stationary materialinside the so-called ‘dead zone’ . A fully-developed pass-ive stress state therefore applies only to hoppers or hop-per-like areas of convergent flow during discharge.

The determination of these pressures is a well-knownproblem that has long been studied. The first methodproposed by Janssen [2], known as the slice-elementmethod, basically consists of finding the solution of thedifferential equation corresponding to the vertical equi-librium in a horizontal grain slice of the silo. This prob-lem is not well-posed, needing additional hypotheses inorder to allow for a solution to the problem. The mostusual of these hypotheses are:

� The independence of the vertical pressures withrespect to the horizontal co-ordinates, that is, thepressure distribution depends only on the verticalcoordinate.

� A constant ratio between vertical and horizontalpressures.

With these assumptions, it is possible to solve theproblem very easily, getting an analytical expression ofthe pressure distribution, valid for any silo. Many other

authors [10,4–7,3,11–15] have contributed in differentways to the improvement of this method that finally hasbecome the basement of most national standards. How-ever, they usually consider significant additional safetycoefficients (see ACI [16], Eurocode [17], DIN [18] orAFNOR [19]) due to the inability of the method to takeinto account the important dynamic effects that appearduring discharge.

Due to its intrinsic hypotheses, the slice-elementmethod accurately estimates the wall pressure, but thehypothesis of constant vertical stress is incorrect. Inorder to get some additional insight, Jenike and Johanson[4–7] included a yield criterion for the steady state flow.The two resulting partial differential equations were con-verted into four ordinary differential equations, whichwere solved by the method of characteristics [20]. Thisapproach can be applied to filling and discharge pro-cesses, although its main disadvantage is the difficultyof extending it to complex geometries, constitutivebehaviours or interactions between walls and grain.

More recently, the possibility of applying numericalmethods to obtain approximate solutions of very com-plex problems has pushed the scientists to look at thisold challenge from a new point of view. For instance,the use of finite elements [21] allows the solution of thedynamic problem in discharging processes for arbitrarygeometries and elaborated constitutive models, simulat-ing the plastic and viscous behaviour of the grain solidand including the contact interaction between grain andsilo walls.

Finite element dynamic studies of the discharge pro-cess in silos have been performed by Kolymbas [22], orRombach [23]. Ruckenbrod and Eibl [24] successfullyused the constitutive models of Lade [25] and the morecomplex mathematical model of Kolymbas [22] todescribe silo discharge and flow patterns by using a vis-coelastoplastic constitutive relationship.

Ooi et al. [26,27] also obtained predictions of thepressure laws in the hopper and studied the influence ofwall imperfections considering elastic behaviour of thebulk material. Rong, Ooi and Rotter [28], Holst, Ooi,Rotter and Rong [29,30] have extensively worked infilling and discharging processes in silos. In filling statesthey employed elastic constitutive behaviour, while indischarging cases a Mohr–Coulomb criterion was used.

Meng, Jofriet and Negi [31,32] proposed a specialclass of secant constitutive relationship for isotropic andcohesionless granular material. They employed an ela-sto-plastic relationship with a Drucker–Prager yield cri-terion and a Prandtl–Reuss flow rule. A simple finiteelement model served as benchmark for the correlationwith experimental results, and a parametric study wascarried out.

Wie�ckowski [33] employs a particular Arbitrary Lag-rangian–Eulerian formulation [34,35] to overcome theproblem of mesh distortion. He also uses a Drucker–

1563M.A. Martınez et al. / Engineering Structures 24 (2002) 1561–1574

Prager yield condition and a non-associative flow ruleto describe the mechanical behaviour of the bulkmaterial and an explicit algorithm to integrate the result-ant equations with respect to time.

Despite all of these works, there still remain importantdifficulties in getting accurate finite element solutions,caused by the extreme distortion of the finite elementmeshes near the silo outlet or the correct calibration ofthe constitutive model of bulk solid.

In order to overcome the intrinsic difficulties ofmethods like the Finite Element Method (FEM) basedon continuum assumptions, the discrete element method(DEM) (Cundall [36] among many others) has recentlybeen applied to this problem. This method treats thegrain mechanical behaviour at the individual particlelevel, and although most researchers use spherical par-ticles, some work on non-spherical particles has alsobeen undertaken [37]. In the DEM, Newton’s equation ofmotion for each single particle replaces the differentialequilibrium equations of continuum mechanics, and themodel describing the particle contacts replaces theconstitutive continuum equations.

Developed during the last ten years, its actual poten-tial of practical application is still unknown, since itdepends strongly on the available computer power,being, for the moment, constrained to the analysis ofsmall models. Some researchers who have employed thismethod to estimate pressure laws in silos have beenYoshida [38] and Rong, Ooi and Rotter [28].

Some interesting papers have been published by Holst,Ooi, Rotter and Rong [29,30] and Sanad, Ooi, Holst andRotter [39] consisting of a summary of several inde-pendent calculations made by researchers from differentcountries. A standard problem of a silo filling and dis-charging was defined and two methods employed: thefinite element and discrete element method. Both com-mercial finite element codes and specifically written pro-grams were used and very different constitutive relation-ships were applied (purely elastic, Drucker–Prager withand without dilatant flow, hyperelastic relations, etc.).Pressure distributions were compared with classicaltheories. A recent state of the art can be found in a spe-cial issue of the Journal of Engineering Mechanics con-cerning “The statics and flow of dense granular sys-tems” [40].

In this paper, we focus on three of the main aspectsof the application of finite elements to the dischargingprocess in silos: the constitutive behaviour (both elasticand perfect elastoplastic models have been compared),the contact between the stored solid and silo walls(friction contact elements have been employed) and thesolution of the mesh distortion problem (a remeshing–rezoning algorithm has been adopted for the first stagesof discharge). A reference example is solved using axi-symmetric finite elements, considering both static anddynamic situations, discussing the influence of the

constitutive law and the flexibility of the wall on thepressure distribution and overpressure dynamic factors.Finally, these results are compared with the pressurelaws proposed by different standards.

2. Finite element modelling of the simulation of thedischarging process

As was pointed out in the previous section, the mostimportant limitations in the application of the FEM tothe dynamic simulation of the discharging process inmetallic silos are: 1. The use of an appropriate constitut-ive law for the stored grain; 2. The excessive mesh dis-tortion near the hopper; 3. The contact interactionbetween the bulk solid and the silo walls. In this sectionwe focus on these three problems, since the fundamen-tals of the finite element method on non-linear problemsare considered well known.

2.1. Bulk solid constitutive behaviour

The first problem concerning the bulk solid constitut-ive relation is essential in the discharging process simul-ation and the determination of the pressure distribution.Two different bulk solid behaviours have been con-sidered here: a purely linear elastic constitutive law anda more complex elastoplastic constitutive model includ-ing a yield or fracture Drucker–Prager criterion usuallyused in soil mechanics and granular materials [9]. Theyield surface expression for the linear Drucker-Pragercriterion is:

F � t�ptgb�d � 0 (1)

Fig. 2 depicts the representation of the linear Drucker–Prager criterion with isotropic hardening for a planestrain state, with

Fig. 2. Graphic representation of the plastic Drucker–Prager cri-terion.


p � s1 � s3 t � s1�s3 (2)

The relationships between the cohesion c and theinternal friction angle f of the classic Mohr–Coulombcriterion with the parameters d and b of the Drucker–Prager criterion (for the extreme cases of associate flowand non-associate flow with null dilatant angle y) arewritten as:

Associate flow (3)

tanb ��3sinf

�1 � 1 /3sin2f

dc

��3cosf

�1 � 1/3sin2f,

Non�associate flow (4)

tanb � �3sinfdc

� �3cosf.

For granular materials, non-associate flow with y �b is usually used. In this work, we have considered non-

dilatant flow (y � 0), that is, incompressible inelasticstrain. Although grain solids are essentially cohesionless,a small value of c must be used to avoid numericalsingularities. Relation between cohesion in Drucker–Prager criterion d and compression yield stress in thehardening law is defined as:

d � �1�13tanb�sc (5)

2.2. Mesh distortion and rezoning algorithm

A total Lagrangian formulation has been used to per-form the simulations. In this approach, due to the largestrains appearing, the elements can become so distortedthat the results are useless. This is exactly what happensin the simulation of discharging processes in silos, dueto the high velocity of the grain exiting the cone, in com-parison to the velocity of the inside grain, leading toimportant local distortions of the mesh near the outlet.

A possibility to solve this problem is to remeshlocally, projecting the results from the original to thenew mesh. This projection approach is known in thespecialised literature as rezoning (ABAQUS, [41]) Theanalysis process, therefore, consists of several steps, i.e.1. Running the analysis with a specific mesh until itbecomes distorted, then 2. The results are mapped ontoa new non-distorted mesh, and 3. The calculation con-tinues with this new mesh. This rezoning procedure maybe done as many times as needed during the analysis.

The rezoning is a projection technique composed ofthree steps, as detailed in the following:

� The values of the element variables are extrapolatedfrom the integration points to the nodes of thedeformed old mesh, by performing some kind ofpatch recovery technique.

� Each integration point and node of the new mesh isdetected to belong to an element of the deformed oldmesh. Therefore, it is necessary that the integrationpoints and nodes of the new mesh are located withinthe boundary defined by the old mesh.

� The values of the variables at the integration points(element variables) and nodes (nodal variables) of thenew mesh are obtained by direct interpolation of thenodal values of the deformed old element to whichthey belong.

The accuracy of the rezoning technique highlydepends on the distortion of the initial mesh and the sizeof the elements. If the initial mesh is not distortedexcessively and if the meshes are fine enough, thisrezoning approach works well. Otherwise, a disconti-nuity on the solution is clearly detected.

The generation of the new mesh may be done manu-ally, by using the mesh generation capabilities of thecode used or, alternatively, an external automatic algor-ithm may be used. In this case, an ‘ in house’ code thatautomatically generates the new mesh and the input filefor ABAQUS has been implemented.

This code reads the geometry of the deformed oldmesh, the nodal and element variables at the time stepdecided by the user, and other data such as groups ofelements and nodes, applied loads, boundary conditions,computation method, etc. It also reads the data of thenew mesh that, in our case, is a subset of a fixed meshfilling all the space in which there will be grain duringdischarging. This fixed mesh will be denoted as ‘patternmesh’ (Fig. 3). Once the new mesh has been definedinside the current boundary, it is written on theABAQUS input file, together with all the data neededfor the rezoning, continuing the simulation automati-cally.

As has been explained already, the new mesh is theparticular region of the pattern mesh placed inside theboundary of the deformed mesh. Therefore, the locationof the nodes of the new mesh are exactly the same asthe ones of the pattern except those close to the boundarythat have to be adapted in order to get coincidencebetween the edges of the new elements with the cur-rent boundary.

The process therefore may be summarised as follows:

� The data of the pattern mesh are obtained from anABAQUS output file.

� The data of the deformed mesh, at the time incrementdecided by the user, and other necessary data are readfrom different ABAQUS output files.

� A polyline is generated using the nodes of the bound-ary as vertexes. The black mesh in Fig. 3a representsthe pattern mesh and the grey mesh the deformedmesh, with the thicker line defining the currentdeformed boundary polyline.


Fig. 3. Automatic creation of the new mesh.

� The nodes closest to each vertex of the polyline aretranslated to the location of the vertex, as shown inFig. 3b. If a node is closest to two different vertexes,the node is translated to the boundary point such thatthe adjacent elements become less distorted.

� All the nodes of the pattern mesh are classifiedaccording to their location in the initial mesh as‘ inside’ , ‘outside’ or ‘on top of the polyline’ .

� Then the elements of the pattern mesh are classifiedaccording to their nodes. A three digit number isassigned to each element: the first digit correspondsto the number of nodes that are inside the polyline,the second to the number of nodes that are located onthe line and the third one to the number of nodes out-side the boundary. Several examples may be observedin Fig. 3.

� All the elements that have nodes inside and outsidethe boundary are modified. In Fig. 3c, for instance,the element on the lower left corner, whose type is211, is transformed into an element type 220 by trans-lating one of its nodes to the polyline (see Fig. 3d).In Fig. 3d, all the elements with type 121 are transfor-med into triangular elements type 120 (see Fig. 3e).

� At this moment there is no element with nodes insideand outside, so the external elements are removed andthe internal elements are associated to the new mesh(Fig. 3f).

All the elements involved in the rezoning process havebeen considered to have the same geometric type andthe same number of nodes, for instance, bilinear four-noded elements. As is clear in Fig. 3f, the new mesh iscomposed of elements with three and four nodes. How-ever, triangular elements are defined as degeneratedfour-noded elements.

The properties of the nodes and elements are mappedto the new nodes and elements of the deformed meshby suitable interpolation. For instance, if a node of thenew mesh is at the same location as a node of thedeformed mesh, belonging to a certain group, this newnode will belong to the same group. In the same way,if the node is placed between two nodes of the deformedmesh belonging to the same group, this intermediatenode is also assigned to the group. Once all the elements,nodes, materials and geometry groups of the new meshhave been defined, it is possible to generate automati-cally the ABAQUS input file.

The final new mesh obtained has the same quality asthe pattern mesh (except near the current boundary,which coincides exactly with the one of the deformedold meshes), then the analysis may continue after rezon-ing (see Fig. 4 for an example of this remeshingalgorithm).

2.3. Contact interaction with the silo wall

The interaction between the stored solid and the silowalls has been modelled with contact surfaces, avoiding,therefore, the penetration of the grain nodes into the wallsurface. All the simulations have been performed con-sidering finite strains and displacements, that is, a fullygeometrically non-linear approach.

2.4. Finite element model

A silo with usual geometric dimensions shown in Fig.5a has been chosen as reference. Wheat has been con-sidered as the bulk material.

We have considered an axisymmetric model com-posed by 840 bilinear quadrilateral axi-symmetrical solidelements with complete integration, CAX4 in the

Fig. 4. Different meshes of the bulk solid at the silo outlet beforeand after applying remeshing algorithm.


Fig. 5. Geometry of the silo.

ABAQUS nomenclature, for the stored grain and rigid(RAX1) or flexible axisymetric shell elements (SAX1)with two nodes for the silo wall depending on theexample. The finite element mesh has been progressivelyrefined near the outlet zone, as is clearly shown inFig. 5b.

Four different meshes have been employed in thedynamic analysis following the ideas described in theprevious section for remeshing. They are shown in Fig.6, being easy to notice how the deformed bulk solid thatpasses through the outlet is kept inside the mesh, but aswill be proved later it has a negligible influence on therest of the stored solid. The pattern mesh is always main-tained as was stated before, allowing us to get accurateresults during the evolution of the discharging process.

For the elastic behaviour of the bulk solid, averagevalues from different standards were computed for thewheat constitutive parameters. These values were very

Fig. 6. Different meshes employed for the dynamic simulation fort � 0, 0.14, 0.27, 0.40 s.

Table 1Elastic material parameters for bulk material (AFNOR)

Internal angle of friction (fv) 23.80°Friction coefficient (mv) 0.308Material density 835 kg/m3

Young modulus 2.0E+05 N/m2

Poisson coefficient 0.4

close to the ones proposed by the AFNOR standard [19],so, finally, we decided to use the values proposed by thisstandard that are included in Table 1.

From these parameters (the internal friction angle inparticular) the material parameters for the Drucker–Prager criterion were computed using Eq. (4) giving thevalues established in Table 2.

Different simulations have been performed to analysethe influence of different constitutive models (elastic vselastic–plastic), the flexibility of the silo walls or the dif-ferences between the results associated to static anddynamic situations. The different examples that we willconsider are:

� Static analyses with linear elastic and elastic–plasticbehaviour of the bulk solid, both with rigid and flex-ible silo walls and both under filling and discharging.

� Dynamic simulation of the discharging process con-sidering elastic–plastic behaviour or failure behaviourof the bulk solid.

3. Results of the static filling analyses

3.1. Elastic behaviour and rigid silo wall

In this first case, a static geometrically non-linearanalysis has been performed, considering initially purelyelastic properties for the grain, using the values includedin Table 1. The silo wall was initially considered as rigidand the outlet closed. Contact surfaces have beenincluded along the cylindrical body, the hopper and theoutlet. The friction coefficients are also included inTable 1.

Vertical displacements and the horizontal stress distri-bution are shown in Fig. 7a,b. The maximum displace-ment appears near the axis of symmetry and the mini-mum near the silo wall, due to friction. Horizontalstresses are almost constant at each height as predictedby the Janssen theory. Only in the region near the joint

Table 2Plastic behaviour parameters for bulk material

Internal angle of the Drucker– 34.95°Prager criterionYield stress 1.0E+01 N/m2


Fig. 7. Vertical displacements and horizontal stresses in a static analysis with elastic behaviour of the bulk solid (rigid silo wall and closed outlet).

between the cylindrical body and the hopper, is this viol-ated, due to normal discontinuity that strongly changesthe local contact stress distribution.

The pressure distributions along the cylindrical bodyand the hopper and the comparison with the ones pro-posed by the four standards mentioned in the introduc-tion are shown in Fig. 8. From them we can concludethat the finite element prediction in the cylindrical bodyis in close agreement with the one proposed by theAFNOR standard. This is not surprising since this is theonly code that follows the Janssen theory strictly, while

Fig. 8. Horizontal pressure and tangential stress distributions over the silo wall for a static analysis with elastic behaviour of the bulk solid andrigid walls, and comparison with the different standards.

the rest of them (DIN, ACI and ENV) include overpres-sure factors. On the other hand, in the hopper, the finiteelement approximation is not very accurate. Forinstance, the expected limit to a zero value of the press-ure when approaching the outlet is not predicted appro-priately due to the wrong assumption of closed outletthat induces a non-negligible pressure value at this point.

The tangential stress along the walls follows the sametendency. Perfect sliding is obtained along almost thewhole wall, so a constant ratio between the friction stressand the horizontal pressure is obtained. Only the nodes


close to the body-hopper joint remain tied to the silowall. Therefore friction stresses are smaller than the onespredicted by the Janssen theory (AFNOR code).

3.2. Elastic behaviour and flexible silo wall

If we now consider flexible walls, in order to repro-duce a more realistic situation, (standard properties forthe steel of E � 200 GPa and n � 0.3 have been used)the vertical displacements and the horizontal stresses arestrongly modified as expected.

The elastic behaviour of the grain and the flexibilityof the walls produce a high reduction of the pressuredistribution of 70% when compared to the ones pre-viously obtained under the rigid wall hypothesis andtherefore much smaller than the ones proposed by thedifferent standards (Fig. 9a,b). In this case, the contactis lost near the body-hopper joint, leading, therefore, tonull pressure in that region. This fact is due to theinability of the grain to follow the wall displacements,since the grain horizontal displacements are essentiallycaused by the values of the high vertical stresses and thePoisson effect. The consequence is the increase of thevertical pressures inside the bulk solid and the decreaseof the horizontal pressure and tangential stresses alongthe cylindrical body. Complementary, this also inducesa strong increase of the wall pressure and friction stressin the hopper, in order to balance vertical forces.

The reason for these unrealistic results is, no doubt,the elastic behaviour of the bulk solid, since without ahorizontal restriction (due to the wall flexibility) thegrain should tend to form a cone with the natural fric-tional slope angle, while the simulation with elasticbehaviour tends to maintain the initial rectangular form.Therefore it seems necessary to include a more realisticmaterial behaviour in order to avoid this problem.

Fig. 9. Horizontal and tangential stresses for different simulations.

3.3. Plastic behaviour

We have considered a perfectly plastic behaviour witha Drucker–Prager criterion as has been previouslyexplained in section 2. The corresponding results areshown in Fig. 9a,b, being clear now that the differencesbetween the situations corresponding to the rigid andflexible walls are much smaller than in the elastic simul-ation. Only some small waves in the normal pressuredistribution near the body-hopper joint appear for theflexible case, due to the interaction with shell bendingeffects. In this region the need of additional reinforce-ments is well known in order to absorb the local bendingstresses, keeping the membrane state predominant in therest of the structure.

A small pressure reduction of about 5–10% is pro-duced in the flexible hypothesis, due to the use of a smallvalue for the cohesion in order to avoid numerical prob-lems.

Rigid or flexible wall hypotheses show similar resultsexcept near the body-hopper joint where the bendingwaves, already mentioned, appear (Fig. 9a,b). Under thehypothesis of flexible walls, the normal pressures alongthe hopper increase slightly, since they may balance thereduction of a 5% of the vertical stresses and the frictionforces along the cylindrical body. Finally, and since theresults obtained using plastic constitutive behaviour arevery similar for rigid and flexible walls, we will consideronly the first situation, due to its greater simplicity.

Some partial conclusions for the simulation of fillingprocesses employing static analyses are:

� The flexible wall hypothesis must be always usedtogether with a plastic behaviour of the bulk material,in order that the stored grain can follow the wall dis-placements. In this case the obtained pressure distri-


bution is in agreement with the standards and well-known experimental data.

� Shell local bending causes pressure waves near thebody-hopper joint.

� A decrease of the pressure values of about 5% in thecylindrical body is obtained when using flexiblewalls, when compared to the pressures computedunder the rigid wall hypothesis.

� A perfect plastic criterion is a good choice for thebulk material constitutive behaviour. In fact almostthe whole grain region becomes plastic, with themaximum plastic strains located at the symmetry axis,near the silo wall and near the body-hopper joint.Plastic equivalent stress distribution is very similarboth for the static and the dynamic analyses. (Fig.12b).

4. Results of the dynamic discharge analyses

Once the geometric model, the contact interaction andthe bulk material behaviour have been checked in thestatic case, it is possible to accomplish a detaileddynamic analysis of the discharging process. Rigid wallshave been considered in this case, since, as has beenstated, when plastic failure behaviour is considered forthe stored material, the influence of the wall flexibilityis not so important, and the computational cost is con-siderably reduced.

An implicit time integration algorithm with a certainamount of numerical damping to dissipate the undesiredeffects of spurious higher time frequencies, the Hilber–Hughes–Taylor method (HHT) [42], has been employedto integrate the evolution equations. This numerical

Fig. 12. Vertical displacements inside the silo for a time t � 0.08sand equivalent plastic strains for t � 0.14 s.

damping is useful in these types of problems to improvethe algorithmic stability. It is important in this case dueto the intermittent contact of the bulk material with thesilo walls. A value of the damping parameter, a ��0.20, for the HHT algorithm has been chosen in orderto damp out the low frequency response.

Two steps are applied: first a static analysis is perfor-med considering the previously mentioned boundaryconditions corresponding to a closed outlet. The secondstep corresponds to the dynamic problem, which appearsafter removing the outlet boundary conditions, simulat-ing its instantaneous opening. The first moments are themost important ones reaching a stationary flow very rap-idly. Small time increments have been chosen, accordingto this high velocity problem, so a time step of �t �3.0 × 10�6 has been used.

The analysis goes on until the mesh distortion is toohigh to get useful results, then the remeshing procedure,previously explained, is used and the associated pro-jected results on the new mesh are computed to continuewith the analysis. Four different remeshing intervalshave been analysed defined by three remeshing pro-cesses that, in our case, have been needed at times t �

0.14, 0.27, 0.40 s. The four meshes have been shownin Fig. 6. The distribution of vertical displacements,equivalent plastic strains and vertical stresses arepresented in Fig. 10a,b, and 10c respectively for t �0.15 s, before the first remeshing process.

Some interesting aspects are easily observed in Fig. 10which support the validity of this simulation: the verticaldisplacement contour plot (Fig. 10a) reproduces qualitat-ively well the experimental results obtained by Kvapil[10], known as Kvapil ellipses, shown in Fig. 11. Kvapilconsidered that the moving particles follow two differentmovements: the primary and the secondary. Particlesdisplace downwards only, due to gravity forces in theprimary movement, while particles inside the secondarymovement region have vertical and horizontal displace-ments. These movement regions have elliptical forms.This secondary movement is extremely important nearthe outlet appearing in every type of silo, while the pri-mary movement arises only in very tall silos.

When the grain begins to flow out of the silo throughthe outlet, both ellipses start to grow and when the pri-mary ellipse reaches the free surface a repose zone isformed. The secondary ellipse grows until arriving at thedeepest point of the repose zone (Fig. 11b), then it startsto decrease. Finally, a discharging funnel is formed thatconstitutes the interface between the slow and the fastmoving areas.

Fig. 10a shows how the FE simulation reproduces theformation of the first Kvapil ellipse during the initialdischarging process. At this time, the maximum verticaldisplacement through the outlet is about 0.13 m, whichrepresents an average velocity at this region of about1 m/s.


Fig. 10. Distribution of vertical displacements, equivalent plastic strains and vertical stresses in the outlet region at t � 0.15 s before remeshing.

Fig. 11. Ellipses of movement for the bulk solid proposed by Kvapil.

The equivalent plastic strain distribution (Fig. 10b)shows that the maximum plastic strains appear in theregion near the outlet, representing the local failure ofthe stored grain, due to its inability to support tensionstresses. The maximum plastic strain component appearsalong the vertical axis just above the outlet. Thisinability to support tension stresses is also observed inthe vertical stress distribution (Fig. 10c); the tensionstresses in almost all this area are equal to the limit of10 N/m2 that has been established as the tension failurestress for this material (Table 2).

The immediate consequence is that the bulk solid thathas passed through the outlet does not affect the restof the grain inside the silo. This is the reason why thecorresponding finite elements are not removed from thecurrent mesh. On the contrary, they are retained in orderto make the boundaries of the two meshes coincident(constituting the current deformed boundary polyline),which is a condition needed in ABAQUS to perform therezoning process.

A portion of the grain remains tied to the hopper, for-

ming a slope angle of about 60° (Fig. 12a). This regioncorresponds to the funnel flow in the discharging processas it is described by the AFNOR standard for a silo ofthese dimensions. Most elements have arrived at thestate of failure at t � 0.14 s (Fig. 12b). The maximumvalues of the plastic strains appear in the hopper regiondue to the tension stresses as has been explained before.There also appear high plastic strains in the cylindricalbody near the wall due to the high friction forces.

In Fig. 13a normal wall pressure distribution is shownfor t � 0.02 s, just after outlet opening. The best-fitcurve to Janssen’s theory is also shown in Fig. 13a withthe values of the associated coefficients. A least meansquare method has been used for the cylinder zone.Results are compared with the static FEM simulation.The dynamic overpressure is shown in Fig. 13b.

The dynamic nature of the simulation leads to stresswaves due to the contact-impact between the grain andthe rigid walls. The high-frequency contribution of thewaves is damped by the numerical damping supplied bythe integration algorithm and by the smoothing producedby the rezoning algorithm.

The distribution is similar both for the dynamic andstatic simulations over the upper part of the silo, whileat levels below y � 6 m, the increase becomesimportant, from 15% to 50% in the hopper-cylinder tran-sition. The dynamic overpressure on the hopper wallreaches values close to 60%, showing a tendency to zeropressure near the outlet. Obviously, in this zone thedynamic overpressure coefficient is negative becausestatic FEM does not show this zero tendency.

Similar results for a further instant, t � 0.50 s, areshown in Fig. 14a,b. The wall pressure distribution iscompared with the best-fit Janssen’s curve and with thestatic simulation. Clearly, the most critical zone is the


Fig. 13. Horizontal pressure and dynamic overpressure coefficient for a time t � 0.02 s compared with the static pressure distribution.

Fig. 14. Horizontal pressure and dynamic overpressure coefficient for a time t � 0.50 s compared with the static pressure distribution.

cylinder body-hopper transition, where an increase of85% is reached, both in cylinder and hopper zone. Asmall increase of the pressure occurs from the initialtime, t � 0.02 s to t � 0.5 s, but these values remainpractically constant until a significant reduction of thegrain level is produced. The normal pressure distributionat levels above y � 6 m does not present this increase.

These results are in agreement with previous refer-ences in the literature Rotter et al. [43] and Sanad et al.[39], using the discrete element and finite elementmethods. Specifically in the FEM simulations a pressureincrease appears in the bottom of silo cylinder body.Unfortunately, a deeper comparison is not possible dueto the different silo dimensions and the presence of ahopper in the example of this paper.

The represented wall friction coefficient was com-puted as:

m �tangencial stress

normal stress, (6)

and is shown in Fig. 15a,b for the static and the dynamicanalyses for t � 0.02 s and t � 0.50 s respectively. Inthe static case the wall developed friction coefficientcoincides with the input wall friction, m � 0.308, (Table

2) for the cylinder zone, which means that perfect slidingoccurred. However this ratio is reduced to m � 0.18 atthe hopper, so bulk material remains tied to the hopperwall (Fig. 7a) and tangential contact stresses are lowerthan the product of normal stresses and input frictiondata.

This tendency is completely different for the dynamicanalysis. For t � 0.02 s, the developed wall frictioncoefficient is higher on the top of the cylinder and in theoutlet, because bulk material begins to go out, while mis lower just in the cylinder-hopper transition, wheregrain remains fixed. This zone is called ‘dead zone’ byAFNOR in the case of mixed discharging. For t �0.50 s, m coincides with input data on the hopper due

to the discharging movement of the material and, whilethis friction ratio is practically constant in the cylinderm � 0.10, where the stored material remains attached tothe wall. Logically there is a point where the slip-stickcondition changes, and this point rises when the dis-charging process advances [4–7].

These distributions show a behaviour very similar tothe one observed by Jenike and Johanson [4–7], inwhose results peak pressure appears when materialchanges from an active to a passive state. This peak of


Fig. 15. Developed wall friction coefficient (tangential-normal stress ratio) for static and dynamic (t � 0.02 s and t � 0.50 s) F.E. predictions.

normal pressure appears at h � 1 m for t � 0.2 s andat h � 2 m for t � 0.50 m (Figs. 13a and 14a) approxi-mately. This change from an active to a passive state isproduced a little above the beginning of slipping in thebulk material (Fig. 15a,b), and its mission is to compen-sate this active–passive stress change. This peak risesalong the silo wall when grain begins to slip.

A comparison between the numerical normal pressuredistributions and the ones predicted by national stan-dards is included in Fig. 16. Firstly in the cylinder bodythe static simulation does not provide accurate results,while the dynamic simulation shows a behaviour equalto AFNOR for t � 0.02 and between AFNOR and DIN-ENV for t � 0.50 s. The pressure distribution predicted

Fig. 16. Horizontal pressure distributions over the silo wall for astatic and dynamic analysis (t � 0.02 s and t � 0.50 s), and compari-son with the different standards.

by ACI standard is too high. The ‘safety coefficients’used by ACI applied to the static prediction of theJanssen theory are 1.5 and 1.7 for ‘dynamic’ and ‘ live’loads respectively, with a total multiplicative coefficientof 2.55. ACI is, therefore, the most conservative of thefour standards studied, even when ACI considers a lowerratio between the horizontal and vertical stresses, which,in theory, should lead to smaller pressures along the silobody (i.e. ACI uses a ratio of 0.42 and AFNOR 0.60).

The behaviour of the FEM simulations on the hopper(Fig. 16) is again near the DIN standard or even to ENVstandard, while AFNOR and ACI do not always estimatepressure on the safety side. However the dynamic simul-ation shows a tendency to zero pressure just above theoutlet, while DIN and ENV are more conservative andpredict non zero pressure on the outlet. It is remarkablethat both standards always predict wall normal pressurehigher than the static or dynamic simulation, though theprocedure of calculation is different. The DIN standardconsiders overpressure coefficients during the discharg-ing process, while the ENV incorporates local pressureconcentration factors in every region.

5. Conclusions

The computation of the pressures acting on metallicsilos, due to the effect of the grain during the dischargingprocess, is one of the most important aspects to be takeninto account in the design of silo structures and the mainobjective of this paper. Numerical simulations have beenmade using the finite element method with some specificfeatures, regarding the bulk solid constitutive behaviour,the treatment of the contact between the stored solid andthe silo walls or the remeshing and rezoning processesneeded to solve the problem of excessive mesh distortionduring discharge.

The first problem addressed has been the simulationof the filling process under the Janssen hypotheses thatare the basis of most European and American codes.


They consist of a static calculation, considering a rigidsilo wall assuming a Coulomb friction model betweenthe grain and the wall. The results obtained with our FEanalysis are in close agreement with the AFNOR stan-dard estimations, while the ones of the DIN, Eurocodeand ACI are always on the safety side, showing thatthese last three standards impose too high safety coef-ficients.

Studying the influence of the flexibility of the wall, itis concluded that the initial model, which assumes a per-fect elastic behaviour of the grain, is not valid. Thepressure along some areas of the wall is reduced toapproximately zero, implying that the grain is not ableto follow the horizontal displacements of the wall. Tosolve this problem a Drucker–Prager perfectly plasticconstitutive model, that is, a failure criterion with nullcohesion, has been employed and compared to the elasticone. The pressure distribution coincides again with theone given by the standards, but, due to the flexibility ofthe wall, its value is reduced by about 5%. In addition,the pressure distributions exhibit some waves, especiallynear the joint between the cylindrical body and the hop-per.

A dynamic analysis has been performed using therigid wall and plastic behaviour for the grain as modelproperties in order to simulate the discharging process.As different authors have commented before, this workconfirms that the most unfavourable time appears duringthe first steps of the discharge. The obtained pressuresare substantially higher than the static ones in somezones of the silo. The dynamic pressure increase is about10–30% for the cylinder zone and reaches 85% in thecylinder–hopper transition, while just above the outlet itcan be negative, due to the non-zero tendency of thestatic pressure distribution. These values are close to theones established by many authors by experimentalmethods [4–7] or by numerical techniques [39,43]. Thedeveloped wall friction coefficient for the dynamicanalysis clearly shows the zones where grain remainsfixed to the wall or the zones of relative movement. Apeak of normal pressure accompanies this change of slip-stick condition, as it is explained in several papers [4–7] whose mission is to compensate active–passive stresschange. This peak rises along the silo wall when grainbegins to slip.

A comparison between different standards shows thatAFNOR leads to the best results for the static filling pro-cess, while the rest of the standards are on the safetyside. The DIN and ENV standards provide the best accu-racy in the dynamic simulation of the discharging pro-cess, while AFNOR predicts lower values and ACIhigher ones.

We can finally conclude that complex phenomena likethe discharging of grain in silos, can be analysed bynumerical simulations, although there are two essentialaspects that have to be taken into account: a correct

constitutive model for the grain, including failure and anadequate treatment of the mesh distortion by means ofappropriate algorithms.

Nevertheless, this simulation presents some restric-tions. Firstly, a full 3-D model must substitute the axi-symmetric model to incorporate eccentric effects in thedischarging process. The constitutive assumptions areanother model limitation, i.e. effects like variation of thegrain density during the analysis are not considered, andshould be verified by experimental results. The contactalgorithm must be improved to get a more realistic ratiobetween friction and normal stresses.

The remeshing and rezoning techniques here used areable to reproduce the discharging process although witha high computational cost. Perhaps a meshless methodcan avoid this inconvenience. However the main prob-lem of this simulation is the difficulty of verification ofthe assumptions of the numerical model.

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